All the works mentioned above concentrated on more trackable systems, where foliations of the space exist at least to some extent. The topological structure of many relevant one-dimensional spaces, such as various hyperbolic attractors, including those derived from Anosov, is very different from manifolds, and that feature has proven to be a challenging aspect of questions concerning entropy of homeomorphisms of those spaces. For example, the pseudo-arc that we are dealing with in the present paper is as far from being a foliated space as possible, since all its arc-components are degenerate to a point. Perhaps a better
paradigm example of one-dimensional continua to start with in this aspect are solenoids, which are inverse limits of the unit circle with finite-sheeted covering maps, and which were used as one of the first examples of hyperbolic attractors.
Let \(\Sigma _{\alpha }\)  denote the solenoid obtained as the inverse limit of covering maps of degree \(\alpha =(p_1,p_2,\ldots )\) , where one can assume without loss of generality that the powers \(p_i\)  are prime numbers (Bing [1]} and McCord [2]}). Each homeomorphism \(h\)  of a solenoid \(\Sigma _{\alpha }\)  has the same topological entropy as a certain automorphism from its automorphism group associated uniquely with \(h\) . This result was proven by Kwapisz [3]}. There is also a way to obtain planar arc-like one-dimensional continua from solenoids by taking appropriate quotient maps.
Namely, generalized Knaster continua \(K_{\alpha }\)  can be defined using sequences of primes \(\alpha =(p_1,p_2,\ldots )\)  and are isomorphic to \(\Sigma _{\alpha }/\sim \)  where the equivalence relation identifies points from \(\Sigma _{\alpha }\)  with their inverses.
The corresponding quotient mapping is exactly two-to-one except at one or two points. These generalized Knaster continua can also be represented as inverse limits \(K_{\alpha } = \underleftarrow{\lim }(I , f_{p_i} )\) . Kwapisz [3]} proved that the mapping class group of \(K_{\alpha }\)  is isomorphic to either \(\otimes ^k_{n=1}\mathbb {Z}\)  or \(\otimes ^k_{n=1}\mathbb {Z}\otimes \mathbb {Z}\) , depending on whether 2 does or does not occur infinitely often in the sequence \(\alpha \) . Apart from generalized Knaster continua, another well-studied family of arc-like continua is the
family of tent inverse limit spaces, that arise as attractors of some planar homeomorphisms [5]}, [6]}, [7]}. It was shown that in this family typical parameters give intrinsically complicated continua that are locally far from being a product of Cantor set and open arc [8]}. However, it turns out that these typical parameters do not give rise to homeomorphic spaces. In fact much more is true. A conjecture due to Ingram, which attracted much attention in the last three decades, asserted that
inverse limit spaces of tent maps
with different slopes \(s\in (\sqrt{2},2]\)  are non-homeomorphic. It was proven in [9]} by Barge, Bruin and Štimac.
As a somewhat surprising by-product of its solution (but in agreement with the results described above), it was shown in
[10]} that the mapping class group of \(X\)  is \(\mathbb {Z}\) .
This in turn was used to characterize possible values of the topological entropy of homeomorphisms on these spaces [11]}. According to it, the entropy is always a non-negative integer multiple of the logarithm of the slope of the tent map.
